Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be a continuous function.
Let $Z(f)$ be the zero of $f$.
Prove that $Z(f)$ is closed.
This is one of problems in my mid-term exam. I have used $\epsilon-\delta$ argument and showed that the complement of $Z(f)$ is open. Hence, $Z(f)$ is closed.
The problem is my professor didn't marked my answer sheet, but a teaching assistant marked it and he marked it wrong. He said, "there is a counter example that a complement of an open set is not closed".
I have no words. As far as i know, that is the definition of closed sets. Am i wrong?